In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions.
Rademacher's theorem states that a Lipschitz map f : Rn → Rm is differentiable almost everywhere in Rn; in other words, for almost every x, f is approximately linear in any sufficiently small range of x. If f is a function from a Euclidean space Rn that takes values instead in a metric space X, it doesn't immediately make sense to talk about differentiability since X has no linear structure a priori. Even if you assume that X is a Banach space and ask whether a Fréchet derivative exists almost everywhere, this does not hold. For example, consider the function f : [0,1] → L1([0,1]), mapping the unit interval into the space of integrable functions, defined by f(x) = χ[0,''x''], this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ x ≤ y≤ 1, then
| 1 | |
| |f(x)-f(y)|=\int | |
| 0 |
|\chi[0,x](t)-\chi[0,y](t)|dt=
| y | |
| \int | |
| x |
dt=|x-y|,
but one can verify that limh→0(f(x + h) - f(x))/h does not converge to an L1 function for any x in [0,1], so it is not differentiable anywhere.
However, if you look at Rademacher's theorem as a statement about how a Lipschitz function stabilizes as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of f instead of its linear properties.
A substitute for a derivative of f:Rn → X is the metric differential of f at a point z in Rn which is a function on Rn defined by the limit
MD(f,z)(x)=\limr →
| dX(f(z+rx),f(z)) | |
| r |
A theorem due to Bernd Kirchheim[1] states that a Rademacher theorem in terms of metric differentials holds: for almost every z in Rn, MD(f, z) is a seminorm and
dX(f(x),f(y))-MD(f,z)(x-y)=o(|x-z|+|y-z|).
The little-o notation employed here means that, at values very close to z, the function f is approximately an isometry from Rn with respect to the seminorm MD(f, z) into the metric space X.